orthogonality – In a Hilbert Space: Minimal $Leftrightarrow$ Biorthogonal

Perhaps obvious, but I’m struggling a bit with (i) $Rightarrow$ (ii) only.

Proposition. Let ${x_n}_{n in mathbb{N}}$ be a sequence in a Hilbert space $H$. Then the following are equivalent:

i) Let $S_n := overline{text{span}{x_n : n neq m}}$. For each $m in mathbb{N}$, $x_m notin S_n$.

ii) Exists ${y_n}_{n in mathbb{N}}$ in $H$ that is biorthogonal to $x_m$, i.e. the inner product is $1$ when $n = m$ and $0$ otherwise.


My idea: I suppose if we let $S_n^{perp}$ be the orthogonal complement of $S_n$, then for any $y_n in S_n^{perp}$, we have $langle x_m, y_n rangle = 0$. But when $n = m$, how can I be sure that $langle x_m, y_n rangle = 1$?

general topology – Any space $X$ can be embedded as a dense subset of a Lindelöf space.

I have the following statement to prove, but I’m not sure how to go about doing so.

Any space $X$ can be embedded as a dense subset of a Lindelöf space.

There is a hint associated to this exercise,

[Hint: Adjoin a point $p$ to $X$ whose nbhds are the sets ${p}cup E$, where $E$ is a subset of $X$ whose complement in $X$ is Lindelöf.]

Linux VM showing incorrect Space in OS

I have create one VM guest (on esxi) , OS Ubuntu 16 server. I assigned it 900 GB but I am seeing only 200 GB of disk space.

root@XXX-log:/temp# df -h
Filesystem                         Size  Used Avail Use% Mounted on
udev                               7.8G     0  7.8G   0% /dev
tmpfs                              1.6G  900K  1.6G   1% /run
/dev/mapper/ubuntu--vg-ubuntu--lv  196G   92G   94G  50% /
tmpfs                              7.9G  8.0K  7.9G   1% /dev/shm
tmpfs                              5.0M     0  5.0M   0% /run/lock
tmpfs                              7.9G     0  7.9G   0% /sys/fs/cgroup
/dev/sda2                          976M  146M  764M  16% /boot
tmpfs                              1.6G     0  1.6G   0% /run/user/0
root@XXX-log:/temp#
root@XXX-log:/temp#
root@XXX-log:/temp#
root@XXX-log:/temp# fdisk -l
Disk /dev/sda: 920 GiB, 987842478080 bytes, 1929379840 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: gpt
Disk identifier: 2C824CE0-94E5-4515-B28D-8FA40983CFF5

Device       Start        End    Sectors  Size Type
/dev/sda1     2048       4095       2048    1M BIOS boot
/dev/sda2     4096    2101247    2097152    1G Linux filesystem
/dev/sda3  2101248 1929377791 1927276544  919G Linux filesystem


Disk /dev/mapper/ubuntu--vg-ubuntu--lv: 200 GiB, 214748364800 bytes, 419430400 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes

What seems wrong here & how to rectify? Kindly advise

functional analysis – contraction in five dimensional Euclidean space

$T(u,v,w,x,y)=(v,0.5w,x,y,u)$

$S(u,v,w,x,y)=(y,u,v,0.5w,x)$

Could any one tell me whether S and T are contraction map on any matrix norm/induced vector norm? Are their composition both way i.e ST and TS, contraction? What can we say about Contraction of SSSSTTTT or TTTTSSSS? Thank you for your response.

storage – Unisphere dynamic pools. Does this mean I can lose more than 1 disk with a raid 5 configuration and 20 drives or how are spare space extents treated?

So emc has implemented dynamic pools which do away with hot spares and instead uses some drive extents as spare space on each drive.

When a drive fails, the contents are copied into this spare space (is it usable living on this spare space?).

When setting up a traditional system, you would normally have a couple hot spares so if a drive fails, it rebuilds onto these spares thus saving you a failure. (Raid 5 with a couple hot spares allows you to lose 3 drives at seperate times i believe).

With this dynamic pool, how does this work? If I have a 20 drive system and use raid 5 on a dynamic pool…what number of drives can I lose (at different times of course) and still be operational?

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I wanted to test making the objects in the world teleport around the player, but I will have problems with the fact that each object is a different size, and teleporting them around will for sure create problems with moving objects. (Big obstacles could be teleported away and not collide with, for example, bullets until the bullet gets teleported too but inside the big obstacle)
Are there other ways to accomplish this?

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disk space – Oracle: Decreasing PCTFREE on large table

Question about Oracle Database Enterprise Edition version 19.10.0.0.0 running as a standalone instance.

I have a table with billions of rows. The table has always had PCTFREE set to 10.

It turns out that we only ever perform inserts on this table, no updates, no deletions. I would like to save disk space by lowering PCTFREE to 0.

As I understand it, if I just do ALTER TABLE x SET PCTFREE 0 then in our case, only future allocated blocks are affected. But all the millions of existing blocks will never be filled above 90% because we never update or delete those rows.

Is there a way to get all the old blocks “re-evaluated” and put back on the freelist after I lower PCTFREE?

I would prefer not to move all of the data (e.g. use CTAS, or expdp+impdp, or delete+insert every row, or dbms_redefinition) because the table is so big that this would take forever and/or use enormous amounts of intermediate space. I just need Oracle to know that it’s now okay to put more rows in the old blocks. Is this possible?